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## Abstract

Numerical simulations of stochastic coalescence in a parcel framework are presented using a series of distribution functions. The equations governing the distribution parameter tendencies are derived using a variational approach with constraints. Solutions with two and three log-normal distribution functions are compared with a conventional benchmark model and the distribution model is shown to produce accurate solutions. Although only coalescence is considered within this paper, the procedures for including further physical processes is discussed. All of the simulations presented use the log-normal distribution although the method is general enough that it could be adapted to use other distributions such as the gamma distribution.

A decrease in the number of dependent variables by as much as by a factor of 10 as well as an equivalent reduction in computation time required for the treatment of the coalescence equation makes the distribution model attractive for multi-dimensional cloud model simulations. Further research in the direction of extending the distribution model for such purposes is currently in progress.

## Abstract

Numerical simulations of stochastic coalescence in a parcel framework are presented using a series of distribution functions. The equations governing the distribution parameter tendencies are derived using a variational approach with constraints. Solutions with two and three log-normal distribution functions are compared with a conventional benchmark model and the distribution model is shown to produce accurate solutions. Although only coalescence is considered within this paper, the procedures for including further physical processes is discussed. All of the simulations presented use the log-normal distribution although the method is general enough that it could be adapted to use other distributions such as the gamma distribution.

A decrease in the number of dependent variables by as much as by a factor of 10 as well as an equivalent reduction in computation time required for the treatment of the coalescence equation makes the distribution model attractive for multi-dimensional cloud model simulations. Further research in the direction of extending the distribution model for such purposes is currently in progress.

## Abstract

A theoretical study has been carried out to determine the relevant microphysical processes which control the survival distance of ice particles failing from cirrus clouds in subsaturated air, and to determine the atmospheric conditions which are necessary for such particles to “seed” lower level supercooled clouds and thereby initiate glaciation. Differential equations were developed which describe the heat and mass transfer during the evaporation of cirrus ice particles. In these equations forced convection and kinetic effects were included. Spherical, columnar and plate-like ice particles were considered. The effect of radiative heat exchange between an ice particle and its environment was studied in terms of maximum and minimum physical limits for the upward and downward radiation fluxes. Using these limits and the known emission and absorption Properties of ice, we concluded that radiative heat transfer changes the survival distance of columnar ice crystals falling from cirrus clouds by less than 10% if the relative humidity of the environmental air is less than 70%. Considering the radiative effects and a wide range of values for the initial size and ice particle bulk density, and for the temperature and humidity conditions of the ambient air, the present theoretical model showed that ice particles could survive distances of up to 2 km when the relative humidity with respect to ice was below 70% in a typical mid-latitude atmosphere. Larger survival distances are only possible if the ambient air has relative humidities larger than 70%. The theoretical model is compared to several field observations on evaporating cirrus ice particles. Good agreement was found with observational data when the atmospheric temperature and humidity profiles were available for the site at which the ice particles were sampled.

## Abstract

A theoretical study has been carried out to determine the relevant microphysical processes which control the survival distance of ice particles failing from cirrus clouds in subsaturated air, and to determine the atmospheric conditions which are necessary for such particles to “seed” lower level supercooled clouds and thereby initiate glaciation. Differential equations were developed which describe the heat and mass transfer during the evaporation of cirrus ice particles. In these equations forced convection and kinetic effects were included. Spherical, columnar and plate-like ice particles were considered. The effect of radiative heat exchange between an ice particle and its environment was studied in terms of maximum and minimum physical limits for the upward and downward radiation fluxes. Using these limits and the known emission and absorption Properties of ice, we concluded that radiative heat transfer changes the survival distance of columnar ice crystals falling from cirrus clouds by less than 10% if the relative humidity of the environmental air is less than 70%. Considering the radiative effects and a wide range of values for the initial size and ice particle bulk density, and for the temperature and humidity conditions of the ambient air, the present theoretical model showed that ice particles could survive distances of up to 2 km when the relative humidity with respect to ice was below 70% in a typical mid-latitude atmosphere. Larger survival distances are only possible if the ambient air has relative humidities larger than 70%. The theoretical model is compared to several field observations on evaporating cirrus ice particles. Good agreement was found with observational data when the atmospheric temperature and humidity profiles were available for the site at which the ice particles were sampled.

## Abstract

A three-dimensional numerical model is used to study the effect of small-scale supersaturation fluctuations on the evolving droplet distribution in the first 150 m above cloud base. The primary purpose of this research is to determine whether the irreversible coupling between the thermodynamics and dynamics due to finite phase relaxation time scales τ*s* is sufficient to produce significant small-scale horizontal variations in supersaturation. Thus, the paper is concerned only with this internal source for thermodynamic variability. All other source terms, such as the downgradient flux of the variance of thermodynamic fields, have purposely been neglected.

Lagrangian particle experiments were run in parallel with the basic Eulerian model. The purpose of these experiments is to relax some of the microphysical parameterization assumptions with respect to assumed distribution shape and as a result add credibility to the results of distribution broadening.

Model results of five cases are presented, representing the cloud condensation nuclei characteristics of typical continental and maritime cumulus with mean dissipation rate of −100 cm^{2} s^{−3}. The results show that for a maritime case of *N*≈100 cm^{−3} and *w¯*=0.5 m s^{−1} the standard deviation of the supersaturation is as large as its horizontal mean. The horizontal variability of all thermodynamic fields is shown to increase significantly with τ*s*. The droplet broadening response to this irreversible coupling effect is found to be significant for the larger values of τ*s* in the Eulerian experiments. The Lagrangian particle experiments showed a somewhat reduced but still significant effect.

Although the experiments do show a broadening effect caused by finite values of τ*s*, in no case were we able to show a continual increase in distribution broadening with height as reported from cumulus observations.

## Abstract

A three-dimensional numerical model is used to study the effect of small-scale supersaturation fluctuations on the evolving droplet distribution in the first 150 m above cloud base. The primary purpose of this research is to determine whether the irreversible coupling between the thermodynamics and dynamics due to finite phase relaxation time scales τ*s* is sufficient to produce significant small-scale horizontal variations in supersaturation. Thus, the paper is concerned only with this internal source for thermodynamic variability. All other source terms, such as the downgradient flux of the variance of thermodynamic fields, have purposely been neglected.

Lagrangian particle experiments were run in parallel with the basic Eulerian model. The purpose of these experiments is to relax some of the microphysical parameterization assumptions with respect to assumed distribution shape and as a result add credibility to the results of distribution broadening.

Model results of five cases are presented, representing the cloud condensation nuclei characteristics of typical continental and maritime cumulus with mean dissipation rate of −100 cm^{2} s^{−3}. The results show that for a maritime case of *N*≈100 cm^{−3} and *w¯*=0.5 m s^{−1} the standard deviation of the supersaturation is as large as its horizontal mean. The horizontal variability of all thermodynamic fields is shown to increase significantly with τ*s*. The droplet broadening response to this irreversible coupling effect is found to be significant for the larger values of τ*s* in the Eulerian experiments. The Lagrangian particle experiments showed a somewhat reduced but still significant effect.

Although the experiments do show a broadening effect caused by finite values of τ*s*, in no case were we able to show a continual increase in distribution broadening with height as reported from cumulus observations.

## Abstract

In an effort to bring more realism cloud-radiation calculations, arising-parcel model of cloud microphysics and a 191 waveband model of atmospheric radiation (ATRAD) have been brought to bear on the problem of cloud absorption of solar radiation, with emphasis on the effect of drops greater than 40–50 μm in radius. The earlier conclusions of Welch and others that such large drops can produce cloud absorptivities in excess of 30% have not been substantiated. Instead we find large-drop enhancements of only 0.02–0.04 in cloud and total atmospheric absorptivities. However, several other, more important influences were uncovered: 1) Large drops make it necessary to know the second and third moments of the drop distribution in order to parameterize the shortwave effect of clouds; parameterizations based only on the third moment (liquid water content) do not consider a wide enough range of variation of drop distribution. 2) Large drops cause a precipitous fall in both cloud and planetary albedo if the supply of liquid water is fixed. 3) Large drops enhance the solar greenhouse effect by distributing solar heating more deeply into the cloud. Plots of spectral heating rate reveal that the spectral regions 1.5–1.8 μm and 1.15–1.3 μm are most important for shortwave heating of clouds.

It is suggested that very large drops may also explain the looming “optical depth paradox,” whereby optical depths deduced from measurements of reflected radiation are much smaller than those calculated from measured liquid water profiles.

## Abstract

In an effort to bring more realism cloud-radiation calculations, arising-parcel model of cloud microphysics and a 191 waveband model of atmospheric radiation (ATRAD) have been brought to bear on the problem of cloud absorption of solar radiation, with emphasis on the effect of drops greater than 40–50 μm in radius. The earlier conclusions of Welch and others that such large drops can produce cloud absorptivities in excess of 30% have not been substantiated. Instead we find large-drop enhancements of only 0.02–0.04 in cloud and total atmospheric absorptivities. However, several other, more important influences were uncovered: 1) Large drops make it necessary to know the second and third moments of the drop distribution in order to parameterize the shortwave effect of clouds; parameterizations based only on the third moment (liquid water content) do not consider a wide enough range of variation of drop distribution. 2) Large drops cause a precipitous fall in both cloud and planetary albedo if the supply of liquid water is fixed. 3) Large drops enhance the solar greenhouse effect by distributing solar heating more deeply into the cloud. Plots of spectral heating rate reveal that the spectral regions 1.5–1.8 μm and 1.15–1.3 μm are most important for shortwave heating of clouds.

It is suggested that very large drops may also explain the looming “optical depth paradox,” whereby optical depths deduced from measurements of reflected radiation are much smaller than those calculated from measured liquid water profiles.

## Abstract

A theoretical model is formulated which allows the processes that control the wet deposition of atmospheric pollutants to be included in cloud dynamic models. The model considers the condensation process and the collision-coalescence process which, coupled together, control the fate of atmospheric aerosol particles removed by clouds and precipitation through nucleation scavenging and impaction scavenging. The model was tested by substituting a simple parcel model for the dynamic framework. In this form the model was used to determine the time evolution of the aerosol particle mass scavenged by drops as well as the aerosol particle mass left unactivated in air as “drop-interstitial” aerosol. In the present computation all aerosol particles are assumed to have the same composition. Our study shows for inside cloud scavenging: 1) collision and coalescencence causes among the various drop size categories a redistribution of the scavenged aerosol particles in such a manner that the main aerosol particle mass is always associated with the main water mass, thus ensuring that if a cloud reaches the precipitation stage it will also return to the ground the main aerosol particle mass scavenged by the cloud; 2) although the main aerosol particle mass is contained in the large drops, the mass mixing ratio of the captured aerosol in the cloud water is larger inside smaller drops than inside larger drops, implying that smaller drops are more contaminated than larger ones; 3) through nucleation scavenging the total number concentration of aerosol particles is predicted to become reduced by 48 to 94% depending on the composition of the particles, the reduction being mainly confined to aerosol particles larger than 0.1 μm in radius. This implies that a drop interstitial aerosol exists that consists of a particle population reduced in number concentration by up to 94% and reduced in mass by several orders of magnitude, as compared to the particle concentration outside the cloud. 4) Although the aerosol particle mass scavenged by impaction scavenging cannot completely be neglected in accounting for the total amount of aerosol particle mass scavenged by clouded it is smaller by several orders of magnitude than the aerosol particle mass removed by nucleation scavenging.

## Abstract

A theoretical model is formulated which allows the processes that control the wet deposition of atmospheric pollutants to be included in cloud dynamic models. The model considers the condensation process and the collision-coalescence process which, coupled together, control the fate of atmospheric aerosol particles removed by clouds and precipitation through nucleation scavenging and impaction scavenging. The model was tested by substituting a simple parcel model for the dynamic framework. In this form the model was used to determine the time evolution of the aerosol particle mass scavenged by drops as well as the aerosol particle mass left unactivated in air as “drop-interstitial” aerosol. In the present computation all aerosol particles are assumed to have the same composition. Our study shows for inside cloud scavenging: 1) collision and coalescencence causes among the various drop size categories a redistribution of the scavenged aerosol particles in such a manner that the main aerosol particle mass is always associated with the main water mass, thus ensuring that if a cloud reaches the precipitation stage it will also return to the ground the main aerosol particle mass scavenged by the cloud; 2) although the main aerosol particle mass is contained in the large drops, the mass mixing ratio of the captured aerosol in the cloud water is larger inside smaller drops than inside larger drops, implying that smaller drops are more contaminated than larger ones; 3) through nucleation scavenging the total number concentration of aerosol particles is predicted to become reduced by 48 to 94% depending on the composition of the particles, the reduction being mainly confined to aerosol particles larger than 0.1 μm in radius. This implies that a drop interstitial aerosol exists that consists of a particle population reduced in number concentration by up to 94% and reduced in mass by several orders of magnitude, as compared to the particle concentration outside the cloud. 4) Although the aerosol particle mass scavenged by impaction scavenging cannot completely be neglected in accounting for the total amount of aerosol particle mass scavenged by clouded it is smaller by several orders of magnitude than the aerosol particle mass removed by nucleation scavenging.

## Abstract

Two- and three-dimensional simulations of cloud systems for the period of 1–7 September 1974 in phase III of the Global Atmospheric Research Programme (GARP) Atlantic Tropical Experiment (GATE) are performed using the approach discussed in Part I of this paper. The aim is to reproduce cloud systems over the GATE B-scale sounding array. Comparison is presented between three experiments driven by the same large-scale conditions: (i) a fully three-dimensional experiment, (ii) a two-dimensional experiment that is an east–west section of the three-dimensional case, and (iii) a high-resolution version of the two-dimensional experiment. Differences between two- and three-dimensional frameworks and those related to spatial resolution are analyzed.

The three-dimensional experiment produced a qualitatively realistic organization of convection: nonsquall clusters, a squall line, and scattered convection and transitions between regimes were simulated. The two-dimensional experiments produced convective organization similar to that discussed in Part I. The thermodynamic fields evolved very similarly in all three experiments, although differences between model fields and observations did occur. When averaged over a few hours, surface sensible and latent heat fluxes and surface precipitation evolved very similarly in all three experiments and evaluated well against observations. Model resolution had some effect on the upper-troposheric cloud cover and consequently on the upper-tropospheric temperature tendency due to radiative flux divergence. When compared with the fully three-dimensional results, the two-dimensional simulations produced a much higher temporal variability of domain-averaged quantities.

The results support the notion that, *as long as high-frequency temporal variability is not of primary importance,* low-resolution two-dimensional simulations can be used as realizations of tropical cloud systems in the climate problem and for improving and/or testing cloud parameterizations for large-scale models.

## Abstract

Two- and three-dimensional simulations of cloud systems for the period of 1–7 September 1974 in phase III of the Global Atmospheric Research Programme (GARP) Atlantic Tropical Experiment (GATE) are performed using the approach discussed in Part I of this paper. The aim is to reproduce cloud systems over the GATE B-scale sounding array. Comparison is presented between three experiments driven by the same large-scale conditions: (i) a fully three-dimensional experiment, (ii) a two-dimensional experiment that is an east–west section of the three-dimensional case, and (iii) a high-resolution version of the two-dimensional experiment. Differences between two- and three-dimensional frameworks and those related to spatial resolution are analyzed.

The three-dimensional experiment produced a qualitatively realistic organization of convection: nonsquall clusters, a squall line, and scattered convection and transitions between regimes were simulated. The two-dimensional experiments produced convective organization similar to that discussed in Part I. The thermodynamic fields evolved very similarly in all three experiments, although differences between model fields and observations did occur. When averaged over a few hours, surface sensible and latent heat fluxes and surface precipitation evolved very similarly in all three experiments and evaluated well against observations. Model resolution had some effect on the upper-troposheric cloud cover and consequently on the upper-tropospheric temperature tendency due to radiative flux divergence. When compared with the fully three-dimensional results, the two-dimensional simulations produced a much higher temporal variability of domain-averaged quantities.

The results support the notion that, *as long as high-frequency temporal variability is not of primary importance,* low-resolution two-dimensional simulations can be used as realizations of tropical cloud systems in the climate problem and for improving and/or testing cloud parameterizations for large-scale models.

## Abstract

A two-dimensional cloud-resolving model with a large domain is integrated for 39 days during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) to study the effects of ice phase processes on cloud properties and cloud radiative properties. The ice microphysical parameterization scheme is modified based on microphysical measurements from the Central Equatorial Pacific Experiment. A nonlocal boundary layer diffusion scheme is included to improve the simulation of the surface heat fluxes. The modified ice scheme produces fewer ice clouds during the 39-day simulation. The cloud radiative properties show significant improvement and compare well with various observations. Both the 39-day mean value (202 W m^{−2}) and month-long evolution of outgoing longwave radiative flux from the model are comparable with satellite observations. The 39-day mean surface shortwave cloud forcing is −110 W m^{−2}, consistent with other estimates obtained for TOGA COARE. The 39-day mean values of surface net longwave, shortwave, latent, and sensible fluxes are −46.2, 182.9, −109.9, and −7.8 W m ^{−2}, respectively, in line with the IMET buoy data (−54.6, 178.2, −102.7, and −10.6 W m^{−2}).

The offline radiation calculations as well as the cloud-interactive radiation simulations demonstrate that a doubled effective radius of ice particles and enhanced shortwave cloud absorption strongly affect the radiative flux and cloud radiative forcing but have little impact on the cloud properties. The modeled albedo is sensitive to the effective radius of ice particles and/or the shortwave cloud absorption in the radiation scheme. More complete satellite observations and theoretical studies are required to fully understand the physical processes involved.

The results suggest that the ice microphysical parameterization plays an important role in the long-term simulation of cloud properties and cloud radiative properties. Future field observations should put more weight on the microphysical properties, cloud properties, and high-quality radiative properties in order to further improve the cloud-resolving modeling of cloud systems and the understanding of cloud–radiation interaction.

## Abstract

A two-dimensional cloud-resolving model with a large domain is integrated for 39 days during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) to study the effects of ice phase processes on cloud properties and cloud radiative properties. The ice microphysical parameterization scheme is modified based on microphysical measurements from the Central Equatorial Pacific Experiment. A nonlocal boundary layer diffusion scheme is included to improve the simulation of the surface heat fluxes. The modified ice scheme produces fewer ice clouds during the 39-day simulation. The cloud radiative properties show significant improvement and compare well with various observations. Both the 39-day mean value (202 W m^{−2}) and month-long evolution of outgoing longwave radiative flux from the model are comparable with satellite observations. The 39-day mean surface shortwave cloud forcing is −110 W m^{−2}, consistent with other estimates obtained for TOGA COARE. The 39-day mean values of surface net longwave, shortwave, latent, and sensible fluxes are −46.2, 182.9, −109.9, and −7.8 W m ^{−2}, respectively, in line with the IMET buoy data (−54.6, 178.2, −102.7, and −10.6 W m^{−2}).

The offline radiation calculations as well as the cloud-interactive radiation simulations demonstrate that a doubled effective radius of ice particles and enhanced shortwave cloud absorption strongly affect the radiative flux and cloud radiative forcing but have little impact on the cloud properties. The modeled albedo is sensitive to the effective radius of ice particles and/or the shortwave cloud absorption in the radiation scheme. More complete satellite observations and theoretical studies are required to fully understand the physical processes involved.

The results suggest that the ice microphysical parameterization plays an important role in the long-term simulation of cloud properties and cloud radiative properties. Future field observations should put more weight on the microphysical properties, cloud properties, and high-quality radiative properties in order to further improve the cloud-resolving modeling of cloud systems and the understanding of cloud–radiation interaction.

## Abstract

Four theoretical approaches are presented for quantitatively determining the intensity of the internal circulation and the flow patterns inside and outside liquid water spheres falling at terminal velocity in air. The first approach assumes creeping flow outside and inside a water sphere, the second assumes potential flow outside and inviscid motion inside a water sphere, the third makes use of boundary layer theory, and the fourth approach uses a numerical method to solve the full Navier-Stokes equation of motion inside and outside a water sphere. The theoretical predictions are compared with data obtained from new quantitative wind tunnel experiments on spherical and deformed water drops. The results show that the creeping flow analysis greatly underestimates the strength of the internal velocity while the inviscid flow analysis greatly overestimates it. On the other hand, the results of the boundary layer approach and of the numerical approach agree reasonably well with the experimental data for drops with radii <500 μ. For larger drops the results of the boundary layer approach greatly overestimate the strength of the internal circulation and predict a completely wrong trend of the variation of the internal velocity with drop size, while the numerical results, although somewhat overestimating the circulation strength, predict the trend correctly. Reasonably good agreement is also found between the observed flow patterns inside the drop and those numerically predicted. In two appendices the effect of the internal circulation on drop shape and hydrodynamic drag is discussed.

## Abstract

Four theoretical approaches are presented for quantitatively determining the intensity of the internal circulation and the flow patterns inside and outside liquid water spheres falling at terminal velocity in air. The first approach assumes creeping flow outside and inside a water sphere, the second assumes potential flow outside and inviscid motion inside a water sphere, the third makes use of boundary layer theory, and the fourth approach uses a numerical method to solve the full Navier-Stokes equation of motion inside and outside a water sphere. The theoretical predictions are compared with data obtained from new quantitative wind tunnel experiments on spherical and deformed water drops. The results show that the creeping flow analysis greatly underestimates the strength of the internal velocity while the inviscid flow analysis greatly overestimates it. On the other hand, the results of the boundary layer approach and of the numerical approach agree reasonably well with the experimental data for drops with radii <500 μ. For larger drops the results of the boundary layer approach greatly overestimate the strength of the internal circulation and predict a completely wrong trend of the variation of the internal velocity with drop size, while the numerical results, although somewhat overestimating the circulation strength, predict the trend correctly. Reasonably good agreement is also found between the observed flow patterns inside the drop and those numerically predicted. In two appendices the effect of the internal circulation on drop shape and hydrodynamic drag is discussed.